Test3d05, 5th 'benchmark test' data file for CPO3D
Relativistic motion in a uniform electric field
The set-up is the same as in test3d03.dat (parabolic motion in a uniform field), except that the energy is 1 MeV and the voltage is 1 MV. At the turning point z should be 0.38312, and is found to be 0.38308, an error of 0.01%
The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.
Solving the relativistic equations of motion for the present conditions, the turning point of the ray should occur at x = xm, z = zm, t = tm, where
xm = x_initial + (U/T)*ln(W/(1+T-U)),
zm = z_initial + (1+T-W)/T,
tm = U/(T*c),
T = initial_kinetic_energy/(m0*c**2),
U = sqrt(T+0.5*T**2),
W = sqrt(1+T+0.5*T**2).
For the energy used in this example,
xm = 0.30692 mm,
zm = 0.38312 mm,
tm = 3.354E-9 ms (that is, 3.354 ps).
The test plane has been put at x = 0.30692, and it can be seen from the output file that zm = .38308 (an error of 0.01%) and that tm is correct. It can also be seen from the detailed information on the ray steps, or from the screen plot, that the crossing point is at the turning point of the ray.
For details see the note on integrating relativistic equations of motion.